Descriptive statistics

Here is why examiners love descriptive statistics: they can test your whole grasp of a data set with one word — skew. Get the mean-median-mode order right under skew, and you have banked easy points.

Descriptive statistics summarize a data set (they describe this sample; they do not infer to a population). Start with central tendency. The mean is the arithmetic average — sensitive to every score, so outliers pull it. The median is the middle score — resistant to outliers. The mode is the most frequent score, the only one usable for nominal data. A distribution can be bimodal (two peaks).

Now the classic trap — skew and the order of the three. In a symmetric normal curve, mean = median = mode (all at the center). In a positive (right) skew, a tail of high scores pulls the mean rightward, so mean > median > mode. In a negative (left) skew, a low tail drags the mean down, so mean < median < mode. Mnemonic: the mean chases the tail, and the tail names the skew (tail points right = positive).

Next, variability — how spread out scores are. The range is max minus min (crude, outlier-driven). The variance is the average squared deviation from the mean. The standard deviation (SD) is the square root of the variance — spread in the original units, the workhorse measure. Bigger SD = more spread.

Finally, distribution shape and outliers. Shape covers symmetry, skew, and kurtosis (peakedness). An outlier is an extreme score; it inflates the range, SD, and variance and drags the mean — which is exactly why the median is the preferred center for skewed data like income.

Clinical Pearl: Under skew, the mean gets dragged toward the tail, and the tail names the skew — right tail = positive skew with mean > median > mode; left tail = negative skew with mean < median < mode.